Mathematical analysis for nonlinear partial differential equations with singular solutions

本研究的目的是求解非线性偏微分方程解与时间相关的奇性。我们解决了以下问题：(1)在具有自由边界的肥皂膜振动问题上，我们建立了处理波型自由边界问题的方法；(2)我们利用离散莫尔斯流构造了一个求解体积约束双曲型方程的数值方法；(3)我们构造了一个体积约束双曲型方程的弱解；(4)我们构造了一个带有体积约束的抛物型方程的弱解，并且证明了解的Hoeld连续性。此外，我们还利用离散Morese流给出了最小化算法的并行机的解算器。这非常有效，特别是对于体积约束问题。这对于弱连接的并行机来说也是非常好的，因为它使用了直接的变分原理。最后，我们要对这个项目的所有参与者表示特别的感谢。

The aim of this research was to solve nonlinear Partial Differential Equations whose solution is expected to have singularities depending on time. The candidates of singularities are defects in harmonic mapping, vortex in Ginzburg-Landau problem and free boundaries.We have solved the following problems;(1)On a Soap film vibration with free boundary, we have established the method to treat wave type free boundary problems(2)We developed a numerical method via the discrete Mores flow for volume constraint conditions(3)We constructed a weak solution to a hyperbolic equation with volume constraint(4)We constructed a weak solution to a parabolic equation with volume constraint and showing Hoelder continuity of thesolutionMoreover we have developed solvers for parallel machine with minimizing algorithm via the discrete Morese flows. This works very well especially for volume constraint problems. This is also very nice for a weak connected parallel machines, because it uses direct method of variational principle.Finally, we would like to express pur special thanks to all participants of this project.